Related topics
Crystal lattices, crystal systems, Bravais-lattice, reciprocal lattice, Miller indices, structure factor, atomic scattering factor, Bragg scattering, characteristic X-rays, monochromatization of X-rays, Bragg-Brentano Geometry.
Principle
Polycrystalline powder samples, which crystallize in the three cubic Bravais types, simple, face-centered and body-centered, are irradiated with the radiation from a X-ray tube with a copper anode. A swivelling Geiger-Mueller counter tube detects the radiation that is constructively reflected from the various lattice planes of the crystallites. The Bragg diagrams are automatically recorded. Their evaluation gives the assignment of the Bragg lines to the individual lattice planes, their spacings as well as the lattice constants of the samples, and so also the corresponding Bravais lattice type.
Equipment
P2542101 PHYWE Systeme GmbH & Co. KG © All rights reserved 3
1 / XR 4.0 expert unit / 09057-99
1 / XR 4.0 Goniometer for X-ray unit, 35 kV / 09057-10
1 / XR 4.0 Plug-in module with Cu X-ray tube / 09057-50
1 / Counter tube, type B / 09005-00
1 / Lithium fluoride monocrystal, mounted / 09056-05
1 / Universal crystal holder / 09058-02
1 / Probe holder for powder probes / 09058-09
1 / Diaphragm tube with nickel foil / 09056-03
1 / Ammonium chloride, 250 g / 30024-25
Fig. 1: XR 4.0 expert unit 09057-99
1 / Potassium chloride, 250 g / 30098-251 / Potassium bromide, 100 g / 30258-10
1 / Molybdenum, 100 g / 31767-10
1 / Micro spoon, special steel / 33393-00
1 / Vaseline, 100 g / 30238-10
1 / Mortar and pestle, d = 90 mm / 32603-00
1 / Software X-ray unit, 35 kV / 14414-61
1 / Data cable USB plug type A/B / 14608-00
Recording equipment:
PC, Windows® XP or higher
P2542101 PHYWE Systeme GmbH & Co. KG © All rights reserved 3
Tasks
1. Record the intensity of the Cu-X-rays back scattered by the four cubic crystal powder samples with various Bravais lattice types as a function of the scattering angle.
2. Calculate the lattice plane spacings appropriate to the angular positions of the individual Bragg lines.
3. Assign the Bragg reflections to the respective lattice planes. Determine the lattice constants of the samples and their Bravais lattice types.
4. Determine the number of atoms in the unit cell.
Set-up
Connect the goniometer and the Geiger-Müller counter tube to their respective sockets in the experiment chamber (see the red markings in Fig. 2). The goniometer block with the analyser crystal should be located at the end position on the right-hand side. Fasten the Geiger-Müller counter tube with its holder to the back stop of the guide rails. Do not forget to install the diaphragm in front of the counter tube.
Insert a diaphragm tube with a diameter of 2 mm into the beam outlet of the tube plug-in unit.
Note
Fig. 3: Set-up of the goniometer
Details concerning the operation of the X-ray unit and goniometer as well as information on how to handle the monocrystals can be found in the respective
Fig. 2: Connectors in the experiment chamber
Procedure
- Connect the X-ray unit via the USB cable to the USB port of your computer (the correct port of the X-ray unit is marked in Figure 4).
- Start the “measure” program. A virtual X-ray unit will be displayed on the screen.
Fig. 5: Part of the user interface of the software
- You can control the X-ray unit by clicking the various features on and under the virtual X-ray unit. Alternatively, you can also change the parameters at the real X-ray unit. The program will automatically adopt the settings.
- Click the experiment chamber (see the red marking in Figure 5) to change the parameters for the experiment. Select the parameters as shown in the text box.
- If you click the X-ray tube (see the red marking in Figure 5), you can change the voltage and current of the X-ray tube. Select the parameters as shown in the text box: Anode voltage UA = 35 kV; anode current IA = 1 mA..
- Start the measurement by clicking the red circle:
Fig. 4: Connection of the computer
- After the measurement, the following window appears:
- Select the first item and confirm by clicking OK. The measured values will now be transferred directly to the “measure” software.
- At the end of this manual, you will find a brief introduction to the evaluation of the resulting spectra.
Note
- Never expose the Geiger-Müller counter tube to the primary X-radiation for an extended period of time.
Sample preparation:
The sample must be so finely powdered that no grains can be felt when a little of it is rubbed between finger and thumb. A relatively high sample concentration can be obtained by mixing the powder with a little vaseline. To do this, transfer a small amount of the sample onto a sheet of paper and use a spatula to knead it to a firm paste. To achieve the highest concentration of material as possible, use very little vaseline (a spatula tip of it). Fill the relatively solid sample paste into the specimen for powder samples and smooth it flush. Use the universal crystal holder to hold the specimen.
Calibration of the goniometer with the LiF single-crystal:
Exact angular positions of Debye-Scherrer reflections are only to be expected when the goniometer is correctly adjusted. Should the goniometer be out of adjustment for any reason whatever, this fault can be corrected either manually or by means of the autocalibration function:
- Automatic calibration:
The anode material of the X-ray tube is automatically identified. The crystal must be manually set under “Menu”, “Goniometer”, “Parameter”. For calibration, select “Menu”, “Goniometer”, “Autocalibration”. The device now determines the optimal positions of the crystal and the goniometer to each other and then the positions of the peaks. The display shows the corresponding calibration curves. The newly configurated zero position of the goniometer system is saved even after switch-off of the X-ray unit.
- Manual calibration
The crystal for analysis must be manually brought to the theoretical Bragg angle ϑ (counter tube correspondingly to 2ϑ). Now search for the intensity maximum of the line by iterative turning of the crystal and counter tube by a few ±1/10° around this angular position. Following this and in coupled mode, bring the crystal and counter tube to the particular zero position corrected by the error value and then confirm with “Menu”, “Goniometer” and “Set to zero”.
Theory and Evaluation
When X-rays of wavelength λ strike a mass of lattice planes of a crystal of spacing d at a glancing angle of ϑ, then the reflected rays will only be subject to constructive interference when Bragg’s condition is fulfilled, i.e.:
(1)
Bragg’s condition implies that all of the waves scattered at the atom are in phase and so amplify each other, whereas partial waves that are scattered in directions not fulfilling Bragg’s conditions are of opposite phase and so extinguish each other. A more realistic way of looking at this must, however take the actual phase relationships of all of the partial waves scattered by the atom in a certain direction into consideration. When there are N atoms in a unit cell, then the total amplitude of the X-rays scattered by the cell is described by the structure factor F, which is calculated by summing up the atomic scattering factors f of the individual N atoms, taking their phases into account.
In general, the following is valid for F:
(2)
where h, k, l = Miller indices of the reflecting lattice planes and un, vn, wn are the coordinates of the atoms in fractions of the particular edge lengths of the unit cell.
As F is in general a complex number, the total scattered intensity is described by |Fhkl|2.
A cubic simple unit cell contains only one atom with the coordinates 000. According to equation (2), therefore, the structure factor F for this lattice type is given by:
(3)
This means that F2 is independent of h, k and l and all Bragg reflections can therefore occur.
The unit cell of a cubic face-centered lattice has 4 atoms at 000, ½ ½ 0, ½ 0 ½ and 0 ½ ½ . The unit cell of a cubic body-centered lattice has in comparison only 2 atoms at 000 and ½ ½ ½ ..
When the lattice only consists of one sort of atom, then the following conditions are valid for the structure factor:
fcc Lattice
|F|2 = 16 f2, with h k l only even or only odd
|F|2 = 0, with h k l mixed
bcc Lattice
|F|2 = 4 f2, with (h + k + l) even
|F|2 = 0, with (h + k + l) odd (4)
The situation is somewhat different when a lattice is made up of different sorts of atoms.
When, for example, an fcc lattice consists of the atoms A and B, whereby the A atoms lie at 000, ½ ½ 0, ½ 0 ½ and 0 ½ ½ , and the B atoms at ½ ½ ½ , 0 0 ½ , 0 ½ 0 and ½ 0 0, then the following additional condition is given for the structure factor F:
Etwas anders ist die Situation, wenn ein Gitter aus verschiedenen Atomen aufgebaut ist.
Besteht z. B. ein fcc-Gitter aus den Atomen A und B, wobei die A-Atome bei 000, ½ ½ 0, ½ 0 ½ und 0 ½ ½ liegen und
die B-Atome bei ½ ½ ½ , 0 0 ½ , 0 ½ 0 und ½ 0 0, so folgt daraus für den Strukturfaktor F die zusätzliche Bedingung:
fcc Lattice with atoms A and B:
|F|2 = 16 (fA + fB)2, with (h + k + l) even and
|F|2 = 16 (fA - fB)2, with (h + k + l) odd (5)
In such fcc lattices, when the atomic scattering factors f of the two atoms are almost equal (fA ≈ fB), then 111 reflections will only be very weak, if they occur at all.
For the cubic crystal system, the spacing d of the individual lattice planes with the indices (hkl) is obtained from the quadratic form:
(a = lattice constant) (6)
From this and equation (1), with n = 1, the quadratic Bragg equation is obtained:
(7)
Examination of fcc lattices
Potassium bromide
Fig. 6 shows the Debye-Scherrer spectrum of potassium bromide (KBr).
Fig. 6: Bragg-Cu-Kα and Cu-Kβ-lines of KBr
As no filter is used for the monochromatization of the X-rays, when individual lines are evaluated consideration must be given to the fact that the very intense lines that result from Kα-radiation are accompanied by secondary lines that result from the weaker Kβ radiation. These pairs of lines can be identified by means of equation (1). It is namely approximately true with λ (Kα) = 154.18 pm and λ (Kβ) = 139.22 pm:
Fig. 7: Bragg-diagram of KBr only with Cu-Kα beam (a nickel filter was used here)
(8)
These values correspond to the quotients of the sinq values (Fig. 6) of the pairs of lines 2-1, 4-3, 6-5 and 9-7, showing that the lines 1, 3, 5 and 7 originate from the Cu Kβ radiation.
The correctness of this conclusion can be shown by a control measurement (see Fig. 7) using the diaphragm tube with nickel foil to reduce the intensity of the Kβ radiation. The reflexes in Fig. 6 that were previously assigned to the Kβ lines are no longer to be seen. As the intensity of the Kα- radiation is also somewhat weakened by the Ni foil, the detection of reflexes of weak intensity at larger glancing angles is made difficult when this is used.
The following method for evaluating the spectrum is given as an example, and is representative for that for the spectra of other samples. First determine the sinϑ and sin2ϑ values for each individual reflex from the angle of diffraction ϑ of the particular line. From equation (7), the ratios of the observed sin2ϑ values must be representable by the ratios of the sums of the squares of the three integer numbers (h,k,l).
The ratios of the sin2 values of the individual lines (n) to the sin2 value of the first line (2) are calculated as in column J of Table 1. The numbering in column E relates to the reflex lines indicated in Fig. 6. In column A, all of the possible h,k,l numbers are listed. Columns B, C and D show the individual ratios of the sums of squares of these numbers.
When an attempt is made to allot the indices 100 or 110 to the first reflexes, then no agreement with the ratios of the sin2ϑ values is found. When the index 111 is assigned to the first line, however, then all of the other lines can be assigned hkl index triplets with a certain accuracy.
Only even or odd numbers are now given, no mixed indices hkl triplets. According to this, KBr forms an fcc lattice. The corresponding lattice plane spacings d, calculated using equation (1), are given in column K. Values for the lattice constant a determined from equation (6) are given in column L. Taking both the Kα lines and the Kβ lines into consideration, the mean value of the lattice constant a is found to be:
a = (655.1 ± 2.9) pm; Δ (a) / a < 0.5%;
(literature value: a = 658.0 pm)
On dividing the total mass M of a unit cell by its volume V, the density ρ is given, so that:
mit (9)
where n = the number of atoms or molecules in the unit cell; m = atomic/molecular mass; mA = atomic/molecular weight; N = 6.022 · 1023 = Avogadro’s number.
On entering the appropriate values for KBr, ρ = 2.75 g · cm-3 and mA = 119.01 g, in equation (9), it follows that n = 3.91 ≈ 4, i.e. the unit cell contains 4 atoms.
Table 1: Evaluation of the Kα-Debye-Scherrer lines of KBr
A / B / C / D / E / F / G / H / I / J / K / Lhkl / h2+ k2+l2 / / / Reflex no. / Intensity / ϑ/° / sinϑ / sin2ϑ / / d / pm / a / pm
100 / 1
110 / 2 / 1
111 / 3 / 1,5 / 1 / 2 / w / 11,80 / 0,20449 / 0,04182 / 1,00 / 377,0 / 652,9
200 / 4 / 2 / 1,33 / 4 / vs / 13,72 / 0,23718 / 0,05625 / 1,34 / 325,0 / 650,1
210 / 5 / 2,5 / 1,67
211 / 6 / 3 / 2
220 / 8 / 4 / 2,67 / 6 / vs / 19,46 / 0,33315 / 0,11099 / 2,65 / 231,4 / 654,5
221/300 / 9 / 4,5 / 3
310 / 10 / 5 / 3,33
311 / 11 / 5,5 / 3,67 / 8 / w / 22,95 / 0,38993 / 0,15204 / 3,64 / 197,7 / 655,7
222 / 12 / 6 / 4 / 9 / s / 24,08 / 0,40801 / 0,16647 / 3,98 / 188,9 / 654,5
320 / 13 / 6,5 / 4,33
321 / 14 / 7 / 4,67
400 / 16 / 8 / 5,33 / 11 / s / 27,97 / 0,46901 / 0,21997 / 5,26 / 164,4 / 657,5
410/322 / 17 / 8,5 / 5,67
441/330 / 18 / 9 / 6
331 / 19 / 9,5 / 6,22
420 / 20 / 10 / 6,67 / 12 / s / 31,69 / 0,52532 / 0,27596 / 6,60 / 146,8 / 656,3
421 / 21 / 10,5 / 7
332 / 22 / 11 / 7,33
422 / 24 / 12 / 8 / 13 / s / 35,03 / 0,57401 / 0,32948 / 7,88 / 134,3 / 657,9
500/430 / 25 / 12,5 / 8,33
510/431 / 26 / 13 / 8,67
511/333 / 27 / 13,5 / 9
520/432 / 29 / 14,5 / 9,67
521 / 30 / 15 / 10
440 / 32 / 16 / 10,67 / 14 / vw / 41,61 / 0,66406 / 0,44097 / 10,54 / 116,1 / 656,7
522/441 / 33 / 16,5 / 11
530/433 / 34 / 17 / 11,33
531 / 35 / 17,5 / 11,67 / 15 / w / 44,56 / 0,70166 / 0,49232 / 11,77 / 109,9 / 650,0
600/442 / 36 / 18 / 12
610 / 37 / 18,5 / 12,33
611/532 / 38 / 19 / 12,67
620 / 40 / 20 / 13,33 / 16 / w / 47,86 / 0,74151 / 0,54983 / 13,15 / 104,0 / 657,5
621/540/443 / 41 / 20,5 / 13,67
541 / 42 / 21 / 14
533 / 43 / 21,5 / 14,33
622 / 44 / 22 / 14,67 / 17 / w / 50,91 / 0,77656 / 0,60242 / 14,40 / 99,3 / 658,5
The Kβ lines 1, 3, 5 and 7 that occur in Fig. 6 are evaluated in Table 2.